From large deviations to semidistances of transport and mixing: coherence analysis for finite Lagrangian data

TL;DR

This paper introduces a novel semidistance measure based on large deviations for analyzing transport and mixing in deterministic flows, enabling the identification of coherent sets from finite Lagrangian data.

Contribution

It proposes a new semidistance derived from large deviations for deterministic flows and finite trajectory data, facilitating the detection of coherent regions in complex flows.

Findings

Semidistance can be computed as shortest paths in a time-dependent graph.

Coherent sets correspond to regions of maximal farness in the semidistance.

Method validated on idealized and standard flow examples.

Abstract

One way to analyze complicated non-autonomous flows is through trying to understand their transport behavior. In a quantitative, set-oriented approach to transport and mixing, finite time coherent sets play an important role. These are time-parametrized families of sets with unlikely transport to and from their surroundings under small or vanishing random perturbations of the dynamics. Here we propose, as a measure of transport and mixing for purely advective (i.e., deterministic) flows, (semi)distances that arise under vanishing perturbations in the sense of large deviations. Analogously, for given finite Lagrangian trajectory data we derive a discrete-time and space semidistance that comes from the "best" approximation of the randomly perturbed process conditioned on this limited information of the deterministic flow. It can be computed as shortest path in a graph with time-dependent weights. Furthermore, we argue that coherent sets are regions of maximal farness in terms of transport and mixing, hence they occur as extremal regions on a spanning structure of the state space under this semidistance---in fact, under any distance measure arising from the physical notion of transport. Based on this notion we develop a tool to analyze the state space (or the finite trajectory data at hand) and identify coherent regions. We validate our approach on idealized prototypical examples and well-studied standard cases.